Progress of Theoretical Physics, Vol. 36, No. 6, December 1966
1277
Representation. Mixing and Sum Rules for Magnetic
Moments from Current Commutation Relations
Satoshi
MATSUDA
. Department of Physics, University of Tokyo, Tokyo
(Received September 10, 1966)
The commutation relation satisfied by the electric dipole moment operators is considered
within the representation mixing scheme. By noting that the expectation value of the dipole
moment operator for the nucleon at infinite momentum in the Z direction is the anomal~us
magnetic moment and its transformation property under the algebra U(3) X U(3) is {(8, l)o
+ (1, S)o; Lz ==- ± 1}, we show that. from the commutation relation of the dipole moment operators a sum rule is derived which relates the anomalous magnetic moments of the proton and
neutron with the isoyector charge radius, where the assumption is made that the commutator
is saturated by a few low-lying resonances. If we take into account the contribution of the
second nucleon resonance N**(1512) to the commutator in addition to that of the nucleon
and N*(1238), the sum rule predicts a reasonable value for the isovector rms charge radius.
A detailed analysis of the various sum rules which have recently been derived from the chiral U(3) X U(3) algebra of currents1) indicates that exact
sum rules may be approximated by sums over a few intermediate states which
might fall into a relatively simple reducible representation of the current algebra.
This observation and some difficulties arising from the assignment of the
positive- and negative-helicity states of the nucleon purely to the (6, 3)1/2 and
(3, 6) _112 representations of U(3) X U(3) *) have led to the suggestion2) ~hat an
appreciable amount of mixing is present.
Foil owing this suggestion, it has been shown3) that the positive-helicity state
of the nucleon can be properly described as having components in the
{(6,3)I 12 Lz=O}, {(3*,3)1/2Lz=O} and {(3, 3*)-1/2Lz=l} representations of
U(3) X U(3), whereas the Jz = 1/2 component of N* (1238, JP = 3/2+) is
purely in the {(6,3)1 12 Lz=O} multiplet:
*l As has been pointed out by various authors, at infinite momentum in the Z direction the
matrix elements of the Z components are equal to those of the time components (both for the vector
and axial-vector currents), so that it is sufficient to discuss an algebra U(3) X U(3), which can then
be identified either as the· chiral or as the collinear current algebra. The irreducible representations
of the latter algebra we shall label by (n, m);. where n and m are the dimensions of the SU(3)
representations generated by Vi+ A~ and V&- A~ respectively; 71. is the eigenvalue of the operator
Ag (we shall refer to this as the "quark helicity"). We define the "orbital helicity" by Lz=Jz-71.
where Jz is the true helicity. (We take the momentum always along the positive Z-axis).
1278
S. Matsuda
( 2)
For the choice of {} = 37°, these assignments lead to the correct predictions of the
experimental values of GA, (the axial-vector coupling constant in the e decay)'
G*, (the strength of the axial-vector transition between the nucleon and the
N* (1238) resonance), and the d/f ratio for the axial-vector current between
the states of baryon octet. Moreover, with the same assumptions and the same
mixing angle one can also predict the ratio .between the anomalous magnetic
moment of the neutron and the strength of the magnetic transition between the
nucleon and N* (1238), if only the property is used that the magnetic-transition .operator transforms under the algebra like {(8; l)o+ (1, 8) 0 Lz= + 1}. The
predicted value for this ratio is in excellent agreement with the experimental
data.
However, as for. the relation between the anomalous magnetic moments of
the proton and neutron, nothing can be said uniquely, since the matrix elements
·of the magnetic transition operator between two nucleons are given by two
independent trans1t10n strengths:
{(6, 3) 112 Lz=0}~{(3*, 3) 112 Lz= -1} and
(3*,3}1 1 2Lz=0}~{(3*,3)1 1 2Lz=-1}. The latter transition is a pure F transition, so that it does not contribute to the neutron moment, but contributes to
the proton moment .. The purpose of this note is to mention further the relation
between the anomalous moments of the proton and neutron in the representation
mixing sheme.
N()w let us consider the electric dipole moment operator
D; =
~d 3xj~(x)x,
(3)
where jb(x) is the time component of the unitary spin current (i = 1, · · ·, 8). It
1
has been noted that the expectation value of Df= ~d 3 x(j~(x) + ,; · j~(x)
3
for the nucleon at infinite momentum along the Z axis is the anomalous magnetic
moment. Under the algebra U(3) X U(3), ·D; transforms as a linear combination of (8, l)o and(l, 8)o while under Jz we have
4)
)x1
[Jz,
1.
e.
D~] =
D~ = ~ (Df +iDD has Lz= +
+ D~,
1 component.
Once the transformation pro-
perties of D; are known, their matrix elements between the nucleons, or the
nucleon and N* (1238) with infinite momentum in the Z direction can easily
be given in terms of .two independent transition strengths: a.....__ {(6, 3)t/2Lz=O}
B{(3*,3) 112 Lz=-1} and ak---.{(3*,3)1/2Lz=0}~{(3*,3)1/2Lz=-1}, using the
Representation Mixing and Sum Rules for Magnetic Moments
1279
representations Eqs. (1) and (2). For later convenience we explicitly write
down the expressions only for the anomalous magnetic moments of the proton
and neutron:
(p,Jz=-
~ liD~jp,Jz=+ ~>
fl.~)(;)
-
~ a cos ()
sin () ( 1 - k tan ())
(4)
and
(n,Jz=-
~ liD~jn,Jz=+ ~>
fl.A (n)
2
· .
ZM - a cosO smO,
3
(5)
where M is the nucleon mass.
From the commutation relation5) of the j~(x, 0) it follows
{6)
where DC±)=D1 +iD2 .. This commutation relation was first used by Cabbibo
and Radicati. 6) It is our aim to show that by using this relation together with
the saturation assumption of the commutator by a few low-lying resonances we
obtain, within the framework of the representation mixing scheme, a sum rule
which connects the anomalous magnetic moments of the proton and neutron
with the isovector charge radius of the nucleon.
If we now take the matrix element of Eq. (6) between the proton states
of infinite momentum along the Z axis, and assume that the commutator on the
left-hand side is approximately saturated by the nucleon and the first nucleon
resonance N* (1238, JP = 3/2+) which are assigned to the representations Eqs.
(1) and (2), we obtain the sum rule 7)
l_(r2 )
3
=
Fl
2
16 2
a cos2 0 sin 2 0(1-_!_tan0) -_!a 2 sin 2 0
9
2
9
'
( 7)
where (r2)p 1 is the isovector charge radius squared (for the form factor F1) of
the nucleon. By eliminating the unknown parameters a and k by use of Eqs.
(4) and (5), we can rewrite Eq. (7) in a more interesting form:
(8)
This relation is ready to be compared with experiment. · An evaluation of the
isovector rms charge radius ( (r2) FJ 112 is obtained by substituting the experimental
values fl.A(p)=l.79 and fl.A(n)=-1.91 and by choosing 0=37°. We thus obtain
S. Matsuda
1280
<r
2)1/2-
which
IS
0. 37
F1 - - -
m1T
to be compared with the experimental value8)
<r 2)1/2
0. 52
m7r
-
F1exp--.- ·
Therefore, the calculated rms charge radius ends up with a value roughly 30%
smaller than the experimental one if we assume that the commutator in Eq. (6)
is saturated ~y N and N* (1238).
In order to improve this situation, let us next take account of some possible
contributions to the commutator in Eq. (6) ·of the second nucleon resonance
3
N**(1512, JP = ; ) , assuming that its Jz= -1/2 state is given by
whereas the Jz=3/2 component is purely in the {(8, l) 312 Lz=O}
choices may be rather reasonable, if we insist on some
interpretation and assign our (3*, 3)1 12 , (3, 3*) -1/ 2 and (8, 1) 312
add the contribution of the N** to the sum rule Eq. (7), we
~ (r
2
)F 1
=
16 2
a sin 2 0 cos2 0(19
~ tanoy- ~ a
2
multiplet. These
SU(6) (SU(6)w)
to 70. 9) If we
obtain10)
sin 2 0
(10)
By usmg Eqs. ( 4) and (5) the sum rule can be transformed irito a more
interesting final expression:
l(r2)F =
1
3
+
(
/-lA(p)- f-lA(n) )
2
1_( f-lA(n)
):!
2M .
__
2M
cos2 0
1
.·( /-lA(p) +3pA(n) )
8 sin 2 0
2M
2
(11)
By substituting the experimental values on the right-hand side, the sum rule
predicts with 0 = 37°
<r 2)1/2-
0. 48
F1 _ __
m7r
which is again to be compared with the experimental value 0.52/m7r. Agreement
with experiment is satisfactory.
Incidentally, in order to predict precisely the experimental value of (r2 )F~ 12
in terms of Eq. (11), we have to choose the mixing angle 0=31 °, which in
Representation Mixing and Sum Rules for Magnetic Moments
1281
turn gives for GA, G*, a=d/f+d and p*/pA(n) 11J,
GA=l.31,
G*=1.14,
a=0.63,
p*jpA(n)=1.65.
We can now summarize the results as follows: By noting that the expectation value of the dipole moment operator . D~ for the nucleon at infinite
momentum in the Z direction is the anomalous magnetic moment and its
transformation property under the algebra i's { (8, l)o+ (1, 8) 0 ; Lz:_ + 1}, we
have shown within the representation mixing scheme that from the commutation
relation satisfied by the dipole moment operators the sum rule is derived which
relates the anomalous magnetic moments of the proton and neutron with the
isovector charge radius squared, where the saturation assumption has been made.
If we take only the nucleon and N* (1238) as the intermediate states of the
commutator, the resulting sum rule is not well satisfied by the experimental
values. On the other hand, if we assume that the nucleon and N* (1238)
together with the second nucleon resonance N** (1512) saturate the commutator,
we obtain a good sum rule which turns out to be fairly well satisfied by the
experimental values.
Recently it has been pointed oue2) that account is to be taken of the
contribution of the non-resonant Eo+ in addition to the resonant M( multipoles in pion photoproduction in order to saturate the sum rule for the nucleon
charge raduis derived by Cabbibo and Radicati 13) and others. 14) In this paper
we have not considered this contribution, and at present, have no means of
incorporating this effect into our sum rule within the framework of the
representation m1xmg.
Acknowledgement
The . author wishes to acknowledge Professor H. Miyazawa for helpful
discussions.
References
1)
2)
3)
4)
5)
6)
7)
M. Gell"Mann, Phys. Rev. 125 (1962), 1067; Physics 1 (1964), 63.
R. F. Dashen and M. Gell-Mann, Proceedings of the Third Coral Gables Conference ·on
Symmetry Pri!ic1Ples at High Energies, University of Miami, 1966 (W. H. Freeman &
Company, San Francisco, California, 1966).
H. Harari, Phys. Rev. Letters 16 (1966), 964; Phys. Rev. Letters 17 (1966), 56.· Lee and
Gernst~in have also obtained similar results using a slightly different mixing model: B. W. Lee
and I. S. Gernstein, Phys. Rev. Letters 16 (1966), 1060.
N. Cabbibo and L. A. Radicati, Phys. Letters 19 (1966), 697.
See reference 1).
See reference 4).
In order to evaluate the sum rule, we also have to specify the representation of the Jz = 3/2.
component of N*, for which we have taken the {(10, 1)3/2Lz=O} multiplet corresponding to the
assignment of its Jz=1/2 component to the {(6, 3)1/2 Lz=O} multiplet (Eq. (2)) (See reference
1282
S. Matsuda
9)). This assignment gives rise to no contribution of the Jz=3/2 component to the sum rule,
since the transition from the {(10, 1)3/2Lz=0} to the other multiplets through Dl is forbidden.
8) L. H. Chan et. al., Phys. Rev. 141 (1966), B 1298.
9) For the decomposition of the states of 56, 70, etc., according to U(3) X U(3), refer to Table 1
in the paper: G. Altarelli, R. Gatto, L. Maiani and G. Preparata, Phys. Rev. Letters 16 (1966),
918.
10) Also in this case the J z = 3/2 component of N**(1512) does not contribute to the sum rule
because of the same reason as in the case of the Jz=3/2 state of N*(1238) (See referenne 7)).
11) See refernce 3).
12) F. J. Gilman and H. J. Schnitzer, Phys. Rev., to be published.
13) See reference 4).
14) S. L. Adler, Phys. Rev. 143 (1966), B 1144.
J. D. Bjorken, Phys. Rev., to be published.
R. F. Dashen and M. Gell-Mann, see reference 2).